CT3 question paper

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INSTITUTE OF ACTUARIES OF INDIA

EXAMINATIONS

10th

May 2011

Subject CT3 Probability & Mathematical Statistics

Time allowed: Three Hours (15.00 18.00)

Total Marks: 100

INSTRUCTIONS TO THE CANDIDATES

1. Please read the instructions on the front page of answer booklet and instructions to

examinees sent along with hall ticket carefully and follow without exception

2. Mark allocations are shown in brackets.

3. Attempt all questions, beginning your answer to each question on a separate sheet.However, answers to objective type questions could be written on the same sheet.

4. In addition to this paper you will be provided with graph paper, if required.

AT THE END OF THE EXAMINATION

Please return your answer book and this question paper to the supervisor separately.


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Q. 1) If ~(0,1) , then for any a > 0, show that

[ > ] [5]

Q. 2) The gross incomes of a random sample of 10 school teachers were recorded (to thenearest 100 and in units of 1,000) and gave the following summary statistics: = 321.6, =10628.31. Income tax is charged at a single rate of 30% on that part ofa persons income above 10,000. Calculate the sample mean and standard deviation of the

10 net incomes.[3]

Q. 3) A uniform random number X divides [0, 1] into two segments. Let R be the ratio of the

smaller versus the larger segment. Compute the density of R.[5]

Q. 4) X has a Poisson distribution with a mean of 2. Y has a geometric distribution on the

integers 0, 1, 2 and also with mean 2. You are given that X and Y are independent.Show that [ = ] =

[5]

Q. 5) You are the consulting actuary to a group of venture capitalists financing a search for

pirate gold.

It is a risky undertaking: With probability 0.80, no treasure will be found and thus the

outcome will be zero.

The rewards are high: With probability 0.20, treasure will be found. The outcome, iftreasure is found, is uniformly distributed to [1000, 5000].

You use the inverse transformation method to simulate the outcome, where large random

numbers from the uniform distribution on [0, 1] corresponds to large outcomes. Yourrandom numbers for the first five trials are 0.95, 0.65, 0.75, 0.55 and 0.85.

Calculate the average of the outcomes of these first five trials.[3]

Q. 6) In order to simulate an observation of a normal random variable it is suggested that

= is used, where X1, X2, Xn is a random sample from a continuous uniform distribution on

the interval (-0.5, 0.5).

(a) Determine the approximate distribution of S. (2)

(b) Determine the value ofn which should be used if S is required to represent a standard

normal random variable. (1)


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(c) Explain why S has the same coefficient of skewness as a standard normal random

variable. (1)

[4]

Q. 7) The random variable X takes values on the interval [0, 2].

You are given:

P( = 1) = 0.25 F( | < 1) = F( | > 1) = 1 E() = 1

Find (1).[5]

Q. 8) If X ~ Poisson() and is large then the approximation

~.(0,1)may be used to derive an approximate confidence interval for .You have observed a single value of 200 from this distribution. Using the aboveapproximation and a continuity correction, derive an approximate 95% symmetrical

confidence interval for .[7]

Q. 9) A random sample of size n is drawn from the distribution with probability density

function

() =

() ; 0 < < , > 0

Derive the asymptotic variance of the maximum likelihood estimator of.[5]

Q. 10) Let X be a random variable distributed in the interval (0, 1) under the density function

( |) = ()()() (1 )Here, (> 0) is some unknown parameter. The gamma function (.) defined by

() = ,

satisfies (a) = (a - 1)! if a is a positive integer.

(a) Derive the mode for the variable X under different permissible values of .

Suppose we wish to test by means of the most powerful test, the following hypothesis

based on a single observation of X:

: =1 : = 2

(3)


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Q. 15) The lengths of 900 sentences randomly sampled from the book The Works, Wealth and

Happiness of Mankind written by H. G. Wells are counted and denoted byX, X, . , X . Here a length of a sentence is defined to be the number of words in thatsentence.

A summary of the data is given by the contingency table below:

Under a proposed model M, ln(X), ln(X), . , ln(X) are independent and identicallydistributed according to (, ).Assuming the above model is correct, define:

(, ) = [ ln() 1 ](, ) = < ln() = 2, 3, , 9(, ) = [ ln() > 5 ](a) Write down the expression for(, ) = 1, 2 , , 10 in terms of standard normal

cumulative distribution function ( ).Suppose you know that = 3 and = 0 . 6 under model M. The following table gives thevalues of expected frequencies E(j) for each of the cells as defined above:

(2)

(b) Complete the above table. (2)

(c) Perform a chi-square goodness-of-fit test to check if the data is indeed a sample

from the model M. Use p-value to draw your conclusions.

Now you are told that you dont know what the values of and are. You look at theactual data and obtain the following estimates =3.0177 and =0.5916. You decide toredo the chi-square test using these estimates. Your revised test statistic value is 11.355.

(4)

(d) Has your conclusion changed from (c) and why? (2)

[10]

ln(X) (-, 1] (1, 1.5] (1.5, 2] (2, 2.5] (2.5, 3]

Freq 1 6 40 121 290

ln(X) (3, 3.5] (3.5, 4] (4, 4.5] (4.5, 5] (5,)

Freq 250 156 34 2 0

j 1 2 3 4 5

E(j) 0.387 5.202 37.125 140.229 *

j 6 7 8 9 10

E(j) * * * * *


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Q. 16) A student after reading up linear regression on a statistics textbook decides to apply the

theory in practice. He computes the average rainfall (y) recorded in his city on each

calendar day (x) during the month of September over the last 10 years.

He wants to fit a linear regression line for y on x and aims to use it to predict rainfall forthe month of September in the next year.

You are given the following:

= ( )( ) =599.62 = ( ) =1,020.60

(a) Calculate the least square estimate of the slope parameter (3)

(b) State the estimate of error variance: and calculate it. (2)

(c) Find

SS, i.e., variability between the responses and their fitted values and so

unexplained by the model. (1)

(d) Define and estimate the coefficient of determination . (1)

(e) Test the significance of the regression coefficient under the hypothesis that there is nolinear relationship between y and x. Comment on your result with the result of

R2 value. (3)

[10]

Q. 17) A psychologist was interested in whether different TV shows lead to more positive

outlook in life. People were split in four groups and then taken to a room to view a

program. The four groups saw: The Tom & Jerry Show, Futurama, The BBC News,No Program. After program a blood sample was taken and serotonin levels (y) measured(more serotonin means more happy).

(a) Calculate the estimates for the:

i. Overall serotonin level

ii. Common underlying variance in serotonin level.

(1)

(3)

(b) Perform an analysis of variance to see if real difference exists between the mean

serotonin levels. State your confidence level. (3)

[7]

**************************

Shows y y2

Tom & Jerry Show 11 7 8 14 11 10 5 66 676

Futurama 4 8 6 11 9 8 46 382

The BBC News 4 3 2 2 3 6 20 78

No Program 7 7 5 4 3 4 4 4 38 196

serotonin levels (y)

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